NCERT Class 9 Maths Solutions Chapter - 10 Circles, Ex - 10.2

Ex - 10.2

Question 1.  Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Solution

A circle is a collection of points which are equidistant from a fix point. This fix point is called as the centre of circle and this equal distance is called as radius of circle. And thus shape of a circle depends on the radius of the circle.
        
So, if we try to superimpose two circles of equal radius, one each other both circles will cover each other.
So, two circles are congruent if they have equal radius.

Now consider two congruent circles having centre O and O' and two chords AB and CD of equal lengths

                         

Now in AOB and CO'D
AB = CD            (chords of same length)
OA = O'C            (radii of congruent circles)
OB = O'D            (radii of congruent circles)
 AOB  CO'D        (SSS congruence rule)

 AOB = CO'D            (by CPCT)
Hence equal chords of congruent circles subtend equal angles at their centres.

Question 2.  Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

Solution

Let us consider two congruent circles (circles of same radius) with centres as O and O'.

In AOB and CO'D
AOB = CO'D        (given)
OA = O'C            (radii of congruent circles)
OB = O'D            (radii of congruent circles)
 AOB CO'D        (SSS congruence rule)
 AB = CD            (by CPCT)
Hence, if chords of congruent circles subtend equal angles at their centres then chords are equal.

Ex-10.1

Ex-10.3